The two leaders will meet in Hanoi on Wednesday and Thursday, eight months after their historic summit in Singapore, the first between a sitting US president and a North Korean leader.
There they pledged to work toward the complete denuclearization of the Korean peninsula, but their vague agreement has produced few results. US Democratic senators and security officials have warned Trump against cutting a deal that would do little to curb North Korea's nuclear ambitions.
As Kim made his way across China by train for the summit, North Korean state media warned Trump on Sunday not to listen to US critics who it said were disrupting efforts to improve ties.
Trump, speaking in Washington on the eve of his departure for Vietnam, said he believed he saw eye to eye with Kim and that they had developed "a very, very good relationship."
"I'm not in a rush. I don't want to rush anybody," Trump said. "I just don't want testing. As long as there's no testing, we're happy."
North Korea conducted its last nuclear test, its sixth, in September 2017. It last tested an intercontinental ballistic missile in November 2017.
The Trump administration has pressed the North to give up the nuclear weapons program, which combined with its missile capabilities, pose a threat to the United States, before it can expect any concessions.
But in recent days, Trump has signaled a possible softening, saying he would love to be able to remove sanctions if there was meaningful progress on denuclearization.
Trump said he and Kim expected to make further progress at the summit and again held out the promise that denuclearization would help North Korea develop its economy.